A New Sixth Order Method for Nonlinear Equations in R

A new iterative method is described for finding the real roots of nonlinear equations in R. Starting with a suitably chosen x0, the method generates a sequence of iterates converging to the root. The convergence analysis is provided to establish its sixth order of convergence. The number of iterations and the total number of function evaluations used to get a simple root are taken as performance measure of our method. The efficacy of the method is tested on a number of numerical examples and the results obtained are summarized in tables. It is observed that our method is superior to Newton’s method and other sixth order methods considered

Solving Polynomial Equations using Modified Super Ostrowski Homotopy Continuation Method

Homotopy continuation methods (HCMs) are now widely used to find the roots of polynomial equations as well as transcendental equations.  HCM can be used to solve the divergence problem as well as starting value problem. Obviously, the divergence problem of traditional methods occurs when a method cannot be operated at the beginning of iteration for some points, known as bad initial guesses. Meanwhile, the starting value problem occurs when the initial guess is far away from the exact solutions.   The starting value problem has been solved using Super Ostrowski homotopy continuation method for the initial guesses between . Nevertheless, Super Ostrowski homotopy continuation method was only used to find out real roots of nonlinear equations.  In this paper, we employ the Modified Super Ostrowski-HCM to solve several real life applications which involves polynomial equations by expanding the range of starting values. The results indicate that the Modified Super Ostrowski-HCM performs better than the standard Super Ostrowski-HCM. In other words, the complex roots of polynomial equations can be found even the starting value is real with this proposed scheme

A new optimal family of three-step methods for efficient finding of a simple root of a nonlinear equation

This study presents a new efficient family of eighth order methods for finding the simple root of nonlinear equation. The new family consists of three steps: the Newton\u27s step, any optimal fourth order iteration scheme and the simply structured third step which improves the convergence order up to at least eight, and ensures the efficiency index 1.6818. For several relevant numerical test functions, the numerical performances confirm the theoretical results